[seqfan] Re: Occupancy Perfect Squares - more cases

[Ron Hardin]

When are even columns perfect squares?  I think I've got it figured out.  First 
the complete list in my /tmp directory of when the even columns are NOT perfect 
squares and when YES the even columns are perfect squares

=== complete list ===
NOT cub stays put or moves to some horizontal or vertical neighbor
NOT cuc moves to some horizontal, vertical or antidiagonal neighbor
NOT cud moves to some king-move neighbor
NOT cug stays put or moves to some horizontal, diagonal or antidiagonal neighbor
NOT cuh stays put or moves to some horizontal or antidiagonal neighbor
NOT cui stays put or moves to some king-move neighbor
NOT cuj stays put or moves to some horizontal, vertical or antidiagonal neighbor
NOT cul moves to some horizontal, vertical or antidiagonal neighbor, without 
2-loops
NOT cun moves to some king-move neighbor, without 2-loops
NOT cup moves to some horizontal, diagonal or antidiagonal neighbor, without 
2-loops
NOT cur moves to some horizontal or antidiagonal neighbor, without 2-loops
NOT cut moves to some horizontal, vertical or antidiagonal neighbor, with no 
occupancy greater than 2
NOT cuu moves to some king-move neighbor, with no occupancy greater than 2
NOT cux stays put or moves to some horizontal or vertical neighbor, with no 
occupancy greater than 2
NOT cuy stays put or moves to some horizontal, vertical or antidiagonal 
neighbor, with no occupancy greater than 2
NOT cuz stays put or moves to some king-move neighbor, with no occupancy greater 
than 2
NOT cva stays put or moves to some horizontal, diagonal or antidiagonal 
neighbor, with no occupancy greater than 2
NOT cvb stays put or moves to some horizontal or anitdiagonal neighbor, with no 
occupancy greater than 2
NOT cvc moves to some horizontal or vertical neighbor, with no 2-loops and with 
no occupancy greater than 2
NOT cvd moves to some horizontal, vertical or antidiagonal neighbor, with no 
2-loops and with no occupancy greater than 2
NOT cve moves to some king-move neighbor, with no 2-loops and with no occupancy 
greater than 2
NOT cvf moves to some horizontal, diagonal or antidiagonal neighbor, with no 
2-loops and with no occupancy greater than 2
NOT cvg moves to some horizontal or antidiagonal neighbor, with no 2-loops and 
with no occupancy greater than 2
NOT cvi moves to some horizontal, vertical or antidiagonal neighbor, with every 
occupancy equal to zero or two
NOT cvj moves to some king-move neighbor, with every occupancy equal to zero or 
two
NOT cvm stays put or moves to some horizontal or vertical neighbor, with every 
occupancy equal to zero or two
NOT cvn stays put or moves to some horizontal, vertical or antidiagonal 
neighbor, with every occupancy equal to zero or two
NOT cvo stays put or moves to some king-move neighbor, with every occupancy 
equal to zero or two
NOT cvp stays put or moves to some horizontal, diagonal or antidiagonal 
neighbor, with every occupancy equal to zero or two
NOT cvq stays put or moves to some horizontal or antidiagonal neighbor, with 
every occupancy equal to zero or two
YES cue moves to some horizontal or antidiagonal neighbor
YES cuf moves to some horizontal, diagonal or antidiagonal neighbor
YES cus moves to some horizontal or vertical neighbor, with no occupancy greater 
than 2
YES cuv moves to some horizontal, diagonal or antidiagonal neighbor, with no 
occupancy greater than 2
YES cuw moves to some horizontal or antidiagonal neighbor, with no occupancy 
greater than 2
YES cvh moves to some horizontal or vertical neighbor, with every occupancy 
equal to zero or two
YES cvk moves to some horizontal, diagonal or antidiagonal neighbor, with every 
occupancy equal to zero or two
YES cvl moves to some horizontal or antidiagonal neighbor, with every occupancy 
equal to zero or two
YES cvr moves to some horizontal or vertical neighbor

Apparently 2-loops omitted and staying put are killers for perfect squares.  
Taking them out, using just the various neighbor arrangements:

=== neighbor arrangements omitting stays-put and 2-loops ===

=== horizontal.or.vertical ===
YES cus moves to some horizontal or vertical neighbor, with no occupancy greater 
than 2
YES cvh moves to some horizontal or vertical neighbor, with every occupancy 
equal to zero or two
YES cvr moves to some horizontal or vertical neighbor

=== horizontal,.vertical.or.antidiagonal ===
NOT cuc moves to some horizontal, vertical or antidiagonal neighbor
NOT cut moves to some horizontal, vertical or antidiagonal neighbor, with no 
occupancy greater than 2
NOT cvi moves to some horizontal, vertical or antidiagonal neighbor, with every 
occupancy equal to zero or two

=== king-move ===
NOT cud moves to some king-move neighbor
NOT cuu moves to some king-move neighbor, with no occupancy greater than 2
NOT cvj moves to some king-move neighbor, with every occupancy equal to zero or 
two

=== horizontal.or.antidiagonal ===
YES cue moves to some horizontal or antidiagonal neighbor
YES cuw moves to some horizontal or antidiagonal neighbor, with no occupancy 
greater than 2
YES cvl moves to some horizontal or antidiagonal neighbor, with every occupancy 
equal to zero or two

=== horizontal,.diagonal.or.antidiagonal ===
YES cuf moves to some horizontal, diagonal or antidiagonal neighbor
YES cuv moves to some horizontal, diagonal or antidiagonal neighbor, with no 
occupancy greater than 2
YES cvk moves to some horizontal, diagonal or antidiagonal neighbor, with every 
occupancy equal to zero or two

King-moves are also killers of perfect squares.

What seems to be going on is that the YES ones can be seen to divide the cells 
into two populations, with isomorphic graphs between them of donor cells (which 
are in population 1) and accumulator cells (which are in population 2), the 
isomorphism reversing the roles.  These two must not interact.  Then the 
accumulation distributions are the same count for each, and the product is a 
perfect square.

Staying put mixes the populations, as do 2-loops being forbidden, and 
king-moves, via interactions.  And so you don't get perfect squares.

 rhhardin at mindspring.com
rhhardin at att.net (either)